There have been two parallel trends of "free speech" discourse over the last several years I have observed with some disdain.
People who complain when other people who do not like what they have said and say so, or otherwise use their private ability to discriminate who may use a platform of speech they have provided, under a mistaken belief that "PC-policing" of this kind is infringing upon their rights to free speech.
Groups of people who cite anecdotal experience to suggest that there are broad, sweeping, and new trends which threaten civil discourse norms of engaging with political opponents in a polite disagreement over facts and motives.
The former is merely stupidity about what the Constitution means; that it restricts the government, but not the people, from imposing restraints upon what someone can say, or what they wish to hear. Which is par for the course, but still really annoying and tedious.
The latter position if true would be a deeply shared concern, but I have not really seen anyone making an attempt to demonstrate that it is in fact what, say, college campuses and college students on said campuses are doing in a new way that they have never before done. These two trends often intermingle in a way that suggests a major problem is in fact the demand for PC behavior. There are points on which I might agree this is the case. But in general, it is not.
The former situation operates in something like this manner: If I am attempting to engage with someone I likely may disagree with, it is best to do so in a method that treats them with a degree of respect. One way to do this is to answer questions in a civil manner. Another is to try to understand their perspective. One can still be quite dismissive of the manner they have arrived at an opinion, of the "facts" they have marshaled in support, of the flimsy nature of logic being used to move from one subject to another, and so on. But one should ultimately try to avoid being dismissive of the person with whom a conversation is being engaged with. This is not always easy to do but it is the proper and civil nature of a debate to carry out, and it has some noble aims in principle behind it. It is possible you might persuade them that they are wrong, or that they have considered a perspective that you have not and that you are in fact in the wrong. At "worst", it may persuade people with whom you haven't even engaged that one of you is offering a more sensible option for interpreting events or their proposed solutions. That is the purpose of engaging in civil debate and discourse with people who disagree with you. What I often find instead is that disagreement is then used as a means of divining and imputing all manner of ill-favored motives, character flaws, ethnic or racial origins (where not otherwise obvious), and an increasing slide into personal attack rather than argumentative cases being constructed for the matter at hand. The effect of such action is to persuade neither person of the wrongness or flaw in their argument, and most likely to make both of you look like complete assholes to any innocent bystander who happens by.
The Internet, for all its flaws in preventing or harnessing the ability of individuals to filter out bullshit from fact, actually does offer some amount of tools for dealing with this as a problem. The ban hammer comes out and away that person goes to cool down for a while, perhaps indefinitely. What it does less well is deal with this when it concerns an entire movement of people condemning an individual or groups of individuals, whereupon they have descended as locusts to feast upon the person, marrow and bone all.
One of the essential difficulties of this is that the former activities, banning an obnoxious troll from your sight and daily interactions, is a subjective action that protects the time and energy of the individual (and sometimes their viewpoints). I find it useful for abusive persons in social media debates, or for people whose arguments are so tedious and pathetic that I no longer wish to expend the time on them. But the latter action, banning a whole host of persons at once, or restricting one or more person's ability to influence said persons through a social media platform entirely, involves a more rigorous analysis than "this person annoys me". And as such, it often passes well into less decorous behavior before anything is or could be done. Threats (true or otherwise), abusive language, bigotry, and so on. Almost all of which is and probably should remain perfectly legal and free from public sanction by the mechanics of government and law. But a platform provided by a private company has no obligation to allow anyone to use it in that way either. It is perfectly legal behavior to be an asshole, and to carry forth senseless mayhem in argumentation. It is also a perfectly sensible action to point out that someone is acting like one.
This is the essential component of what gets termed PC. What is "politically correct"? A simple stab at an answer would be "avoiding being an asshole in public". Expressions of bigotry or intolerance do not need to be tolerated. Merely being annoying or offensive might need to be tolerated at some level, but doesn't need to be sought out either.
This is the part that I am having some difficulty understanding however is how this intersects with college experiences. There are definitely many specific stories indicating that there are some people who have difficulty understanding what to do with academic freedom, or whether academic freedom should exist, and instituting speech codes with fairly arbitrary purposes, and trying to suppress the public speaking appearances of controversial figures. I've encountered some people with a rather poor grasp of freedom of speech in academic settings myself, in debates on public forums, and so on. Getting from there to the supposition that the entire architecture of freedom of inquiry and debate is under assault because "kids today are weak and don't want to engage with tough ideas!" is a very large stretch however.
I consider these concepts like "trigger warnings" to be reasonably robust in the culture at large. We tend to use mechanics like parental ratings systems to screen out material we may not want to view on TV or in films or hear in music. This is not limited to trying to restrict access for children. But to tell ourselves as adults or parents that this is going to have some graphic material and to decide whether or not we want to deal with a show about rape and sexual abuse, or brutal violence or torture. To be sure, I probably have more tolerance with viewing nudity or sexuality, or violence, and certainly with hearing "strong language" than many other people. Nevertheless, I usually want to know that's what I'm about to be in for, implicitly or explicitly rather than just sit down and all the sudden be in the middle of hearing about a brutal rape case or viewing a fictional torture sequence in a film or TV show. If I were taking or teaching an ethics class, I would not expect to dive right into asking about incest as a moral topic. Or abortion. Or even humans (Westerners at least) eating insects (or other humans) as food. There's a certain amount of mental preparation most people need to make in order to get past shock or disgust and try to dig into the content.
If and where these concepts like trigger warnings or safe spaces are being used to destroy discussion altogether, or to prevent controversial figures from speaking and learning what, if any, rationale they might have for their views, or otherwise to present only a sanitized and sanctioned understanding of a topic, I would agree they are being misused and applied to an over broad practice of policing thought and expression. This is not generally what people are doing with them in academic settings. It's more like giving people a minute to collect themselves. I would feel it works more like the slow and relaxing climb up a roller coaster ride, before diving and twisting around.
One of the many beneficial features of free speech and free inquiry is that it allows people to go more or less anywhere with a thought or a hypothesis and to explore many ideas. It also means that when we are treading off into an area of exploration of ideas or the expression of them that is likely to be offensive or troublesome, other people will be free to say so. We can then ignore them. That is likely to make us come off not so well in a social sense, that it will have cultural and social consequences. This should encourage us not to automatically abandon our esoteric and unpopular quest for what seems to be forbidden knowledge. But instead to think harder about its merits and how to express them, or to acknowledge perhaps there are few enough merits to discount the idea more or less entirely and to move on into other inquiries more worth our time. Or at least better forms of expression to get across our ideas. A key value of free inquiry and free speech is the process of ideas having to do some amount of rigorous battle with each other. Rather than simply being able to express something freely without complaint, we should expect that people might marshal their energy to oppose us. And that should lead us to make or seek out the very best and most rigorous arguments for not only what we believe we have learned and discovered to be true about the world around us, but what others believe as well. Such a process should humble us about our own level of certainty. And will also provide some outlets for controversy and debate. Even basic fundamental premises like "major political candidates should not say racist things and be able to get elected" will be tested from time to time, giving us the space to re-evaluate ourselves alongside our fellow citizens, classmates, teachers, friends, family members, and so on. To recognize whether we seem to be progressing or not with difficult ideas like how to deal with issues of racism or sexism. Or crime. Or punishment. Or education.
Even the premises of speech themselves are challenged from time to time. Very strong advocates for its utility should be willing to re-examine the value of speech and understand how robust it is, or how weak it has become as institutional practice if that is indeed the case. As well as understanding the varieties of expression will include shocking and distasteful methods or topics that can be approached more cautiously and perhaps more productively as a result.